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tg2x/x
Trazado el gráfico de la función/ tg2x/x

Gráfico de la función y = tg2x/x

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       tan(2*x)
f(x) = --------
          x
f(x)=tan⁡(2x)xf{\left(x \right)} = \frac{\tan{\left(2 x \right)}}{x} 
f = tan(2*x)/x
Gráfico de la función
02468-8-6-4-2-1010-5050
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
Puntos de cruce con el eje de coordenadas X
El gráfico de la función cruce el eje X con f = 0
o sea hay que resolver la ecuación:
tan⁡(2x)x=0\frac{\tan{\left(2 x \right)}}{x} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=31.4159265358979x_{1} = 31.4159265358979
x2=73.8274273593601x_{2} = 73.8274273593601
x3=70.6858347057703x_{3} = 70.6858347057703
x4=−89.5353906273091x_{4} = -89.5353906273091
x5=53.4070751110265x_{5} = 53.4070751110265
x6=42.4115008234622x_{6} = 42.4115008234622
x7=−64.4026493985908x_{7} = -64.4026493985908
x8=14.1371669411541x_{8} = 14.1371669411541
x9=−1.5707963267949x_{9} = -1.5707963267949
x10=−78.5398163397448x_{10} = -78.5398163397448
x11=−53.4070751110265x_{11} = -53.4070751110265
x12=−67.5442420521806x_{12} = -67.5442420521806
x13=89.5353906273091x_{13} = 89.5353906273091
x14=−9.42477796076938x_{14} = -9.42477796076938
x15=−20.4203522483337x_{15} = -20.4203522483337
x16=28.2743338823081x_{16} = 28.2743338823081
x17=62.8318530717959x_{17} = 62.8318530717959
x18=9.42477796076938x_{18} = 9.42477796076938
x19=−80.1106126665397x_{19} = -80.1106126665397
x20=50.2654824574367x_{20} = 50.2654824574367
x21=36.1283155162826x_{21} = 36.1283155162826
x22=−97.3893722612836x_{22} = -97.3893722612836
x23=95.8185759344887x_{23} = 95.8185759344887
x24=45.553093477052x_{24} = 45.553093477052
x25=−100.530964914873x_{25} = -100.530964914873
x26=92.6769832808989x_{26} = 92.6769832808989
x27=34.5575191894877x_{27} = 34.5575191894877
x28=1.5707963267949x_{28} = 1.5707963267949
x29=−29.845130209103x_{29} = -29.845130209103
x30=65.9734457253857x_{30} = 65.9734457253857
x31=23.5619449019235x_{31} = 23.5619449019235
x32=20.4203522483337x_{32} = 20.4203522483337
x33=78.5398163397448x_{33} = 78.5398163397448
x34=59.6902604182061x_{34} = 59.6902604182061
x35=−47.1238898038469x_{35} = -47.1238898038469
x36=100.530964914873x_{36} = 100.530964914873
x37=−72.2566310325652x_{37} = -72.2566310325652
x38=21.9911485751286x_{38} = 21.9911485751286
x39=7.85398163397448x_{39} = 7.85398163397448
x40=−17.2787595947439x_{40} = -17.2787595947439
x41=84.8230016469244x_{41} = 84.8230016469244
x42=4.71238898038469x_{42} = 4.71238898038469
x43=−37.6991118430775x_{43} = -37.6991118430775
x44=−81.6814089933346x_{44} = -81.6814089933346
x45=−21.9911485751286x_{45} = -21.9911485751286
x46=26.7035375555132x_{46} = 26.7035375555132
x47=64.4026493985908x_{47} = 64.4026493985908
x48=12.5663706143592x_{48} = 12.5663706143592
x49=−87.9645943005142x_{49} = -87.9645943005142
x50=−42.4115008234622x_{50} = -42.4115008234622
x51=−3.14159265358979x_{51} = -3.14159265358979
x52=−14.1371669411541x_{52} = -14.1371669411541
x53=−94.2477796076938x_{53} = -94.2477796076938
x54=−51.8362787842316x_{54} = -51.8362787842316
x55=−69.1150383789755x_{55} = -69.1150383789755
x56=15.707963267949x_{56} = 15.707963267949
x57=18.8495559215388x_{57} = 18.8495559215388
x58=40.8407044966673x_{58} = 40.8407044966673
x59=−43.9822971502571x_{59} = -43.9822971502571
x60=−6.28318530717959x_{60} = -6.28318530717959
x61=58.1194640914112x_{61} = 58.1194640914112
x62=−28.2743338823081x_{62} = -28.2743338823081
x63=48.6946861306418x_{63} = 48.6946861306418
x64=−83.2522053201295x_{64} = -83.2522053201295
x65=−34.5575191894877x_{65} = -34.5575191894877
x66=−95.8185759344887x_{66} = -95.8185759344887
x67=81.6814089933346x_{67} = 81.6814089933346
x68=−75.398223686155x_{68} = -75.398223686155
x69=−36.1283155162826x_{69} = -36.1283155162826
x70=−91.106186954104x_{70} = -91.106186954104
x71=94.2477796076938x_{71} = 94.2477796076938
x72=86.3937979737193x_{72} = 86.3937979737193
x73=−59.6902604182061x_{73} = -59.6902604182061
x74=87.9645943005142x_{74} = 87.9645943005142
x75=−56.5486677646163x_{75} = -56.5486677646163
x76=−15.707963267949x_{76} = -15.707963267949
x77=−23.5619449019235x_{77} = -23.5619449019235
x78=−12.5663706143592x_{78} = -12.5663706143592
x79=−61.261056745001x_{79} = -61.261056745001
x80=−7.85398163397448x_{80} = -7.85398163397448
x81=67.5442420521806x_{81} = 67.5442420521806
x82=80.1106126665397x_{82} = 80.1106126665397
x83=6.28318530717959x_{83} = 6.28318530717959
x84=29.845130209103x_{84} = 29.845130209103
x85=97.3893722612836x_{85} = 97.3893722612836
x86=−50.2654824574367x_{86} = -50.2654824574367
x87=−25.1327412287183x_{87} = -25.1327412287183
x88=−73.8274273593601x_{88} = -73.8274273593601
x89=37.6991118430775x_{89} = 37.6991118430775
x90=−86.3937979737193x_{90} = -86.3937979737193
x91=51.8362787842316x_{91} = 51.8362787842316
x92=43.9822971502571x_{92} = 43.9822971502571
x93=56.5486677646163x_{93} = 56.5486677646163
x94=−45.553093477052x_{94} = -45.553093477052
x95=−65.9734457253857x_{95} = -65.9734457253857
x96=75.398223686155x_{96} = 75.398223686155
x97=−39.2699081698724x_{97} = -39.2699081698724
x98=−31.4159265358979x_{98} = -31.4159265358979
x99=72.2566310325652x_{99} = 72.2566310325652
x100=−58.1194640914112x_{100} = -58.1194640914112
Puntos de cruce con el eje de coordenadas Y
El gráfico cruce el eje Y cuando x es igual a 0:
sustituimos x = 0 en tan(2*x)/x.
tan⁡(0⋅2)0\frac{\tan{\left(0 \cdot 2 \right)}}{0}
Resultado:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- no hay soluciones de la ecuación
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
primera derivada
2tan⁡2(2x)+2x−tan⁡(2x)x2=0\frac{2 \tan^{2}{\left(2 x \right)} + 2}{x} - \frac{\tan{\left(2 x \right)}}{x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−1.05731518790035⋅10−14x_{1} = -1.05731518790035 \cdot 10^{-14}
Signos de extremos en los puntos:
(-1.0573151879003484e-14, 2)


Intervalos de crecimiento y decrecimiento de la función:
Hallemos los intervalos donde la función crece y decrece y también los puntos mínimos y máximos de la función, para lo cual miramos cómo se comporta la función en los extremos con desviación mínima del extremo:
Puntos mínimos de la función:
x1=−1.05731518790035⋅10−14x_{1} = -1.05731518790035 \cdot 10^{-14}
La función no tiene puntos máximos
Decrece en los intervalos
[−1.05731518790035⋅10−14,∞)\left[-1.05731518790035 \cdot 10^{-14}, \infty\right)
Crece en los intervalos
(−∞,−1.05731518790035⋅10−14]\left(-\infty, -1.05731518790035 \cdot 10^{-14}\right]
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(la segunda derivada es igual a cero),
las raíces de la ecuación obtenida serán los puntos de flexión para el gráfico de la función indicado:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
segunda derivada
2(4(tan⁡2(2x)+1)tan⁡(2x)−2(tan⁡2(2x)+1)x+tan⁡(2x)x2)x=0\frac{2 \left(4 \left(\tan^{2}{\left(2 x \right)} + 1\right) \tan{\left(2 x \right)} - \frac{2 \left(\tan^{2}{\left(2 x \right)} + 1\right)}{x} + \frac{\tan{\left(2 x \right)}}{x^{2}}\right)}{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−34.5647499747227x_{1} = -34.5647499747227
x2=−31.4238795850742x_{2} = -31.4238795850742
x3=−1.70345385344701x_{3} = -1.70345385344701
x4=−23.5725441053903x_{4} = -23.5725441053903
x5=−20.4325778560184x_{5} = -20.4325778560184
x6=92.6796806391247x_{6} = 92.6796806391247
x7=45.5585800365993x_{7} = 45.5585800365993
x8=70.6893710694627x_{8} = 70.6893710694627
x9=−53.4117551818158x_{9} = -53.4117551818158
x10=−91.108930812026x_{10} = -91.108930812026
x11=97.3919391186353x_{11} = 97.3919391186353
x12=−80.1137330680957x_{12} = -80.1137330680957
x13=−45.5585800365993x_{13} = -45.5585800365993
x14=72.26009053615x_{14} = 72.26009053615
x15=40.8468237012746x_{15} = 40.8468237012746
x16=43.9879795423184x_{16} = 43.9879795423184
x17=−86.396691473835x_{17} = -86.396691473835
x18=−65.9772346210581x_{18} = -65.9772346210581
x19=−29.8535013062402x_{19} = -29.8535013062402
x20=20.4325778560184x_{20} = 20.4325778560184
x21=−9.45113176509331x_{21} = -9.45113176509331
x22=−36.1352322065599x_{22} = -36.1352322065599
x23=75.401539073973x_{23} = 75.401539073973
x24=73.8308132724072x_{24} = 73.8308132724072
x25=−89.5381826161457x_{25} = -89.5381826161457
x26=−56.5530879290695x_{26} = -56.5530879290695
x27=−7.88551699226233x_{27} = -7.88551699226233
x28=4.7641136505727x_{28} = 4.7641136505727
x29=37.7057405793536x_{29} = 37.7057405793536
x30=−3.21693803233743x_{30} = -3.21693803233743
x31=−6.32240235152853x_{31} = -6.32240235152853
x32=14.1547994988746x_{32} = 14.1547994988746
x33=80.1137330680957x_{33} = 80.1137330680957
x34=−12.5861920129716x_{34} = -12.5861920129716
x35=34.5647499747227x_{35} = 34.5647499747227
x36=12.5861920129716x_{36} = 12.5861920129716
x37=−51.8411006147136x_{37} = -51.8411006147136
x38=−67.5479428567926x_{38} = -67.5479428567926
x39=62.8358313576226x_{39} = 62.8358313576226
x40=−61.2651370068557x_{40} = -61.2651370068557
x41=−94.2504320159027x_{41} = -94.2504320159027
x42=6.32240235152853x_{42} = 6.32240235152853
x43=−28.2831693809014x_{43} = -28.2831693809014
x44=−72.26009053615x_{44} = -72.26009053615
x45=87.9674361388446x_{45} = 87.9674361388446
x46=67.5479428567926x_{46} = 67.5479428567926
x47=−25.1426792428098x_{47} = -25.1426792428098
x48=58.1237648337733x_{48} = 58.1237648337733
x49=−95.8211848661661x_{49} = -95.8211848661661
x50=−14.1547994988746x_{50} = -14.1547994988746
x51=−15.7238413065886x_{51} = -15.7238413065886
x52=48.6998188986395x_{52} = 48.6998188986395
x53=28.2831693809014x_{53} = 28.2831693809014
x54=31.4238795850742x_{54} = 31.4238795850742
x55=−22.0025031048686x_{55} = -22.0025031048686
x56=−39.2762719612422x_{56} = -39.2762719612422
x57=−78.5429991376474x_{57} = -78.5429991376474
x58=50.2704549016424x_{58} = 50.2704549016424
x59=56.5530879290695x_{59} = 56.5530879290695
x60=−43.9879795423184x_{60} = -43.9879795423184
x61=89.5381826161457x_{61} = 89.5381826161457
x62=100.53345156734x_{62} = 100.53345156734
x63=65.9772346210581x_{63} = 65.9772346210581
x64=−75.401539073973x_{64} = -75.401539073973
x65=42.4173935405436x_{65} = 42.4173935405436
x66=7.88551699226233x_{66} = 7.88551699226233
x67=22.0025031048686x_{67} = 22.0025031048686
x68=86.396691473835x_{68} = 86.396691473835
x69=−47.1291935757396x_{69} = -47.1291935757396
x70=94.2504320159027x_{70} = 94.2504320159027
x71=−69.1186550951654x_{71} = -69.1186550951654
x72=15.7238413065886x_{72} = 15.7238413065886
x73=−83.2552079908878x_{73} = -83.2552079908878
x74=−97.3919391186353x_{74} = -97.3919391186353
x75=59.69444802068x_{75} = 59.69444802068
x76=−42.4173935405436x_{76} = -42.4173935405436
x77=23.5725441053903x_{77} = 23.5725441053903
x78=95.8211848661661x_{78} = 95.8211848661661
x79=−59.69444802068x_{79} = -59.69444802068
x80=26.7128919644683x_{80} = 26.7128919644683
x81=−58.1237648337733x_{81} = -58.1237648337733
x82=−100.53345156734x_{82} = -100.53345156734
x83=78.5429991376474x_{83} = 78.5429991376474
x84=−73.8308132724072x_{84} = -73.8308132724072
x85=−81.6844693977618x_{85} = -81.6844693977618
x86=29.8535013062402x_{86} = 29.8535013062402
x87=−87.9674361388446x_{87} = -87.9674361388446
x88=51.8411006147136x_{88} = 51.8411006147136
x89=36.1352322065599x_{89} = 36.1352322065599
x90=9.45113176509331x_{90} = 9.45113176509331
x91=1.70345385344701x_{91} = 1.70345385344701
x92=18.8627971275898x_{92} = 18.8627971275898
x93=−37.7057405793536x_{93} = -37.7057405793536
x94=−50.2704549016424x_{94} = -50.2704549016424
x95=84.825948721768x_{95} = 84.825948721768
x96=81.6844693977618x_{96} = 81.6844693977618
x97=−17.2932000627273x_{97} = -17.2932000627273
x98=−64.4065306806787x_{98} = -64.4065306806787
x99=53.4117551818158x_{99} = 53.4117551818158
x100=64.4065306806787x_{100} = 64.4065306806787
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
x1=0x_{1} = 0

lim⁡x→0−(2(4(tan⁡2(2x)+1)tan⁡(2x)−2(tan⁡2(2x)+1)x+tan⁡(2x)x2)x)=163\lim_{x \to 0^-}\left(\frac{2 \left(4 \left(\tan^{2}{\left(2 x \right)} + 1\right) \tan{\left(2 x \right)} - \frac{2 \left(\tan^{2}{\left(2 x \right)} + 1\right)}{x} + \frac{\tan{\left(2 x \right)}}{x^{2}}\right)}{x}\right) = \frac{16}{3}
lim⁡x→0+(2(4(tan⁡2(2x)+1)tan⁡(2x)−2(tan⁡2(2x)+1)x+tan⁡(2x)x2)x)=163\lim_{x \to 0^+}\left(\frac{2 \left(4 \left(\tan^{2}{\left(2 x \right)} + 1\right) \tan{\left(2 x \right)} - \frac{2 \left(\tan^{2}{\left(2 x \right)} + 1\right)}{x} + \frac{\tan{\left(2 x \right)}}{x^{2}}\right)}{x}\right) = \frac{16}{3}
- los límites son iguales, es decir omitimos el punto correspondiente

Intervalos de convexidad y concavidad:
Hallemos los intervales donde la función es convexa o cóncava, para eso veamos cómo se comporta la función en los puntos de flexiones:
Cóncava en los intervalos
[100.53345156734,∞)\left[100.53345156734, \infty\right)
Convexa en los intervalos
[−1.70345385344701,1.70345385344701]\left[-1.70345385344701, 1.70345385344701\right]
Asíntotas verticales
Hay:
x1=0x_{1} = 0
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=lim⁡x→−∞(tan⁡(2x)x)y = \lim_{x \to -\infty}\left(\frac{\tan{\left(2 x \right)}}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(tan⁡(2x)x)y = \lim_{x \to \infty}\left(\frac{\tan{\left(2 x \right)}}{x}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función tan(2*x)/x, dividida por x con x->+oo y x ->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=xlim⁡x→−∞(tan⁡(2x)x2)y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(2 x \right)}}{x^{2}}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(tan⁡(2x)x2)y = x \lim_{x \to \infty}\left(\frac{\tan{\left(2 x \right)}}{x^{2}}\right)
Paridad e imparidad de la función
Comprobemos si la función es par o impar mediante las relaciones f = f(-x) и f = -f(-x).
Pues, comprobamos:
tan⁡(2x)x=tan⁡(2x)x\frac{\tan{\left(2 x \right)}}{x} = \frac{\tan{\left(2 x \right)}}{x}
- No
tan⁡(2x)x=−tan⁡(2x)x\frac{\tan{\left(2 x \right)}}{x} = - \frac{\tan{\left(2 x \right)}}{x}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = tg2x/x
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